Nonparaxial Mathieu and Weber Accelerating Beams

Peng Zhang,1 Yi Hu,2,3 Tongcang Li,1 Drake Cannan,4 Xiaobo Yin,1,5 Roberto Morandotti,2

Zhigang Chen,3,4 and Xiang Zhang1,5,*1NSF Nanoscale Science and Engineering Center, 3112 Etcheverry Hall, University of California, Berkeley, California 94720, USA

2Institut National de la Recherche Scientifique, Varennes, Quebec J3X 1S2, Canada3TEDA Applied Physics School, Nankai University, Tianjin 300457, China

4Department of Physics and Astronomy, San Francisco State University, San Francisco, California 94132, USA5Materials Science Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA

(Received 29 August 2012; published 7 November 2012)

We demonstrate both theoretically and experimentally nonparaxial Mathieu and Weber accelerating

beams, generalizing the concept of previously found accelerating beams. We show that such beams bend

into large angles along circular, elliptical, or parabolic trajectories but still retain nondiffracting and self-

healing capabilities. The circular nonparaxial accelerating beams can be considered as a special case of

the Mathieu accelerating beams, while an Airy beam is only a special case of the Weber beams at the

paraxial limit. Not only do generalized nonparaxial accelerating beams open up many possibilities of

beam engineering for applications, but the fundamental concept developed here can be applied to other

linear wave systems in nature, ranging from electromagnetic and elastic waves to matter waves.

DOI: 10.1103/PhysRevLett.109.193901 PACS numbers: 42.25.�p, 03.50.De, 41.20.Jb, 41.85.�p

Self-accelerating beams have stimulated growingresearch interest since the concept of Airy wave packetswas introduced from quantum mechanics [1] into optics in2007 [2]. As exact solutions of the paraxial wave equation(which is equivalent to the Schrodinger equation), Airybeams propagate along parabolic trajectories and areendowed with nondiffracting and self-healing properties[2,3]. In the past five years, Airy beams have been studiedintensively both theoretically and experimentally [4].Possible applications of Airy beams have also been pro-posed and demonstrated, including guiding microparticles[5], producing curved plasma channels [6], and dynami-cally routing surface plasmon polaritons [7]. However, it isimportant to point out that Airy beams are inherentlysubjected to the paraxial limit. As such, when a self-accelerating Airy beam moves along a parabola and even-tually bends into a large angle, it will escape its domain ofexistence and finally break down. Recently, researchefforts have been devoted to overcome the paraxial limitof Airy beams, and circular nonparaxial acceleratingbeams (NABs) have been identified theoretically and dem-onstrated experimentally [8–11]. Those NABs, found asexact solutions of the Helmholtz equation (HE) in circularcoordinates, travel along circular trajectories beyond theparaxial limit, in a fundamentally different fashion fromthe well-known Bessel, Mathieu, and parabolic nondif-fracting beams that travel along straight lines [12–14].However, unlike Airy beams, the circular NABs cannotbe simply scaled (by squeezing or stretching the transversecoordinates) to obtain different accelerations, and theirbeam paths have to be predesigned to match only certaincircular trajectories defined by the Bessel functions[10,11]. This naturally brings about a series of fundamental

questions: Can a NAB bend itself along other trajectoriesrather than circles? If so, would such a beam maintain itsnondiffracting and self-healing properties while bending tolarge angles and different paths? Is it possible to find aNAB that can be scaled to control the acceleration, thusleading to beneficial practical implementations?In this Letter, we demonstrate both theoretically and

experimentally nonparaxial Mathieu accelerating beams(MABs) and Weber accelerating beams (WABs), general-izing the concept of previously discovered acceleratingbeams into the full domain of the wave equation. Suchnew families of accelerating beams, found as exact solu-tions of the HE in different coordinate systems without theneed of using the paraxial approximation, bend to largeangles along elliptical and parabolic trajectories whilepreserving their unique nondiffracting and self-healingnature. Furthermore, we show that the circular NABs foundpreviously represent only a special case of the ellipticalMABs, whereas the parabolic WABs represent a perfectcounterpart of the Airy beams but without the paraxiallimit and are absolutely scalable for acceleration control.Experimentally, we demonstrate finite-energy MABs andWABs bending to large angles beyond the paraxial limitand observe their nondiffracting and self-healing propagationin free space. Such new and generalized NABs provide largerdegrees of freedom for launching and controlling the desiredbeam trajectories for practical applications. This approach isapplicable to other linear wave systems in nature, rangingfrom electromagnetic and elastic waves to matter waves.Let us first start with MABs by solving the HE in elliptic

coordinates. Here we seek a one-dimensional MAB MðxÞpropagating along an ellipse in the x-z plane with the twosemiaxes a, b of the ellipse oriented along the transverse

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x and longitudinal z axes, as sketched in Fig. 1(a). Bysetting a < b, the corresponding elliptic coordinates canbe constructed via the identities x ¼ h sinh� sin�, z ¼h cosh� cos�, where � 2 ½0;1Þ and � 2 ½0; 2�Þ are

the radial and angular variables, respectively, and h ¼ja2 � b2j1=2 is the interfocal separation. Assuming thatthe MAB takes the form of Mð�;�Þ ¼ Rð�Þ�ð�Þ, the HEcan be split into the following modified and canonicalMathieu differential equations:

d2Rð�Þd�2

� ð�� 2q cosh2�ÞRð�Þ ¼ 0; (1a)

d2�ð�Þd�2

þ ð�� 2q cos2�Þ�ð�Þ ¼ 0; (1b)

where � is the separation constant, q ¼ k2h2=4 is aparameter related to the ellipticity of the coordinate sys-tem, and k ¼ 2�=� is the wave number (� is the wave-length in the medium). Solutions of Eqs. (1a) and (1b) aregiven by radial and angular Mathieu functions [15]. Toform a MAB, the beam constructed from those solutionsshall travel along an ellipse and preserve the shape withregard to the elliptic coordinate system. For simplicity, weconsider a MAB as

Mð�;�Þ ¼ Rmð�;qÞðcemð�;qÞ � isemð�; qÞÞ; (2)

where Rmð�; qÞ represents a radial Mathieu function,while cemð�; qÞ and semð�;qÞ correspond to the even andthe odd solutions of angular Mathieu functions at the sameorderm, respectively. Note that Eq. (2) describes the typical

optical MABs of our interest, for which m> ð2qÞ1=2 isusually a relatively large number so that the even and oddradial Mathieu functions tend to merge [13,15,16].Similar to the circular NAB cases [10,11], Eq. (2) indi-

cates that a perfect MAB circulates clockwise along anellipsewhile preserving its shape in the elliptical coordinatesystem, thus representing a longitudinal elliptic vortex [16].However, to construct an ideal MAB with infinite energy,both forward and backward propagating components arerequired. For a physically realizable beam emitted from asingle source and spatially self-accelerating while propa-gating along the positive z axis, we take a similar approachused in our previous work [11] to introduce a finite-energyMAB. By transforming back into the Cartesian coordinatesand shifting the main lobe of the beam close to zero, thefinite MAB takes the following form at z ¼ 0:

M1ðxÞ ¼ expð��xÞH�xþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 � 2q

q=k

�

� Rm

�Re

�arccosh

�i

�xþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 � 2q

q=k

�=h

��;q

�;

(3)

where HðxÞ is a Heaviside function, Re means the realpart, and � is a positive real number. Apparently, such abeam contains either a forward or a backward propagatingcomponent along an upper half-ellipse with a semiaxis

a � ðm2 � 2qÞ1=2=k, resulting in a maximum self-bendingof 180�. The Fourier spectrum associated with such aMABwithout the exponential truncation can be determined by

�1ðfxÞ ¼ ðk2 � f2xÞ�1=2 exp

�ifx

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 � 2q

q=k

þ i arctan

�semðarccosðfx=kÞ;qÞcemðarccosðfx=kÞ;qÞ

��; (4)

where fx represents the transverse spatial frequency.By simply interchanging the x and z coordinates, we

can readily construct the MABs for the case a > b, forwhich the beam propagates along the half-ellipses with

a � ðm2 þ 2qÞ1=2=k. In this case, the beam profiles andtheir spectra can be determined by the following equations

M2ðxÞ ¼ expð��xÞH�xþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 þ 2q

q=k

�

� Rm

�arccosh

��xþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 þ 2q

q=k

�=h

�; q

�; (5)

�2ðfxÞ ¼ ðk2 � f2xÞ�1=2 exp

�ifx

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 þ 2q

q=k

þ i arccot

�semðarcsinðfx=kÞ; qÞcemðarcsinðfx=kÞ;qÞ

��: (6)

Equations (4) and (6) provide the mask information for thegeneration of the MABs from the Fourier space [2,3].

FIG. 1 (color online). (a) Illustrations of different trajecto-ries associated to Mathieu accelerating beams (MABs);(b) amplitude of the MABs at a < b (blue, dashed), a ¼ b (black,dotted), and a > b (red, solid) at z ¼ 0; (c) intensity (top) andphase (bottom) of the Fourier spectra of the MABs in (b) as afunction of the normalized spatial frequency (fx=k); (d)–(f) Side-view propagations of the MABs in (b), where the z ¼ 0 position ismarked by the white dashed line. Note that the horizontal andvertical scales are identical in the figures.

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To examine the beam propagation dynamics of the aboveformulized MABs, we perform numerical simulations bysolving the nonparaxial wave equations. The results at � ¼532 nm,m ¼ 80, h ¼ 8:7 �m, � ¼ 104, are juxtaposed inFigs. 1(b)–1(f), where (b)–(c) depict the beam profiles, theintensities, and the phases of the Fourier spectra of theMABs at z ¼ 0 for a < b, a ¼ b, and a > b. Apparently,at a ¼ b (h ¼ 0), the MAB turns into a circular NAB. Theside-view propagations of the three beams are depicted inFigs. 1(d)–1(f), respectively, where all the beams start froma given position on the negative z axis. Note that thehorizontal and vertical scales are identical in the figures.Clearly, the constructed MABs self-bend to large anglesalong elliptical trajectories in contrast to the trajectory of acircular NAB as shown in Fig. 1(e). Although the threebeams at z ¼ 0 are quite similar in both real and Fourierspaces (the intensity of the spectra are identical for all threebeams), their trajectories of propagation and beam diffrac-tion properties are distinctly different. As one can notice,only the case a ¼ b corresponds to a true nondiffractingNAB, while for the other two cases a < b and a > b, thesize of the main lobe of the MABs changes slightly after along propagation. The diffraction property of the MABswill be revisited below.

We point out that Eqs. (4) and (6) contain nontrivialinformation for unraveling the underlying physics of theMABs. One simple fact is that the beam structure is deter-mined by the combination of m and q. This means that, at acertain desired beam width, one can set the beam into differ-ent trajectories, resulting in a larger freedom for beam engi-neering in comparison to circular NABs. Importantly, underthe condition q ! 0 or m ! þ1, Eqs. (4) and (6) tend tomerge and eventually turn into Eq. (5) in [11] for the circularNABs except for a trivial phase constant, as illustrated inFig. 1(c). Under such a condition, a ! b, and consequentlythe elliptic trajectory of the beam gradually changes into acircle [see Fig. 1(e)]. This clearly indicates that the circularNABs demonstrated recently in [10,11] represent a specialcase of the MABs presented in this work. In addition, wemention that Eqs. (4) and (6) derived here only involve thesolutions of Eq. (1b), which indicates that one can experi-mentally construct MABs solely with angular Mathieu func-tions, without the need of solving the high order modifiedMathieu equations.

Next, following a similar procedure for the MABs, letus solve the HE for the WABs in parabolic coordinateswith z ¼ �, x ¼ ð2 � �2Þ=2, 2 ð�1;þ1Þ, and � 2½0;1Þ. Assuming that the WAB takes the form ofWð�; Þ ¼ �ð�Þ#ðÞ, the HE can be split into [14,15]

d2�ð�Þd�2

þ ðk2�2 þ 2kÞ�ð�Þ ¼ 0; (7a)

d2#ðÞd2

þ ðk22 � 2kÞ#ðÞ ¼ 0; (7b)

where 2k is the separation constant. The solutions ofEqs. (7a) and (7b) can be determined by the same Weber

differential equation (also called parabolic cylinder differ-ential equation) but the corresponding eigenvalues haveopposite signs [14,15]. To avoid complex notations, weset > 0 and construct a WAB propagating towards thepositive z axis as

Wð�;Þ ¼ Wpð�;ÞðWeð;�Þ � iWoð;�ÞÞ; (8)

where Wp can be either an odd or an even solution of

Eq. (7a), while We and Wo are the even and odd solutionsof Eq. (7b), respectively. The Fourier spectrum associatedwith such a WAB reads as

’ðfxÞ ¼ expðifx=kþ i lnðtanðarccosðfx=kÞ=2ÞÞÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 � f2x

p : (9)

By taking fx=k � 1, one can readily prove that Eq. (9)turns into expð�if3x=3k

3Þ=k, representing the typicalFourier spectrum of an Airy beam. This clearly indicatesthat the Airy beam is the paraxial approximation of ourWAB. In addition, we point out that, although such a WABcomes from the exact solution of the HE and has infiniteenergy, it contains only a forward or a backward propaga-tion component, different from the MABs and circularNABs. Therefore, a physically realizable finite-energyWAB can be established solely by introducing an expo-nential aperture. By transforming back into the Cartesiancoordinates, such a truncated WAB takes the form at z ¼ 0

WðxÞ ¼ expð��xÞWpðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixþ =k

q;Þ: (10)

From Eq. (9), one can readily deduce the mask informationneeded for generating the WAB using the Fourier trans-formation method.Figure 2 shows a typical example of our WABs at � ¼

532 nm, ¼ 40, and � ¼ 104 and its comparison with anAiry beam with a similar beam size, where (a)–(d) depictthe beam profiles, the phases of the Fourier spectra [inten-sity is the same as that in Fig. 1(c)], and the side-viewpropagations of the WAB and the Airy beam, while (e)illustrates the evolution of the beam widths over propaga-tion for different beams. From Figs. 2(c)–2(e), it is obviousthat, although the WAB and the Airy beams are all sup-posed to follow parabolic trajectories, the Airy beambreaks up quickly when bending to large angles. In starkcontrast, the WAB can maintain their fine features wellbeyond the paraxial regime. Similar to the MABs, althoughthe WABs are shape-preserving solutions in paraboliccoordinates, they are no longer perfectly diffraction freeafter being transformed back to Cartesian coordinates.Nevertheless, the diffraction of the MABs and WABs aremuch weaker and largely suppressed in comparison withthat of a Gaussian beam of the same size, as shown inFig. 2(e). Considering there are no true linear nondiffract-ing beams in reality, it is thus reasonable to consider theMABs and WABs as nondiffracting beams.Let us now compare the trajectories and accelerations

of the WABs and the Airy beams. A WAB determined by

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Eq. (10) follows x ¼ �ðk=4Þz2, while an Airy beamfollows x ¼ �ð1=4k2x03Þz2 with x0 being the normaliza-

tion constant for the coordinate x. As such, a WAB accel-erates along parabolic trajectories determined by the order, similar to an Airy beam. From Eq. (7), one can readilydeduce that the WABs are scalable as well. Under thenormalization of x0, the beam trajectory turns into x ¼�ðkx0=4Þz2, and the corresponding beam profile is deter-mined by Eq. (10) with the solutions of Eq. (7a) recasted bynormalizing k with 1=x0. Thus, one can control the accel-eration of WABs simply by squeezing or expanding thetransverse coordinate x. Apparently, the WABs represent aperfect counterpart of the Airy beams beyond the paraxiallimit. Notice that the phase mask embedded in Eq. (9) doesnot contain any special function. Therefore, many applica-tions proposed for the Airy beams may well be imple-mented with controllable WABs.

To experimentally demonstrate the MABs and WABs,we use the same setup previously used for generatingcircular NABs [11], except that now the holographic masksare reconfigured with both the phase and intensity infor-mation required to generate the desired acceleratingbeams. Specifically, a laser beam (� ¼ 532 nm) modulatedby a spatial light modulator is sent into a Nikon 60�water-immersion objective lens (NA ¼ 1:0, f ¼ 2:3 mm) to syn-thesize the beams. To directly visualize the beam path, ananosuspension containing 50 nm polystyrene beads isused to scatter the light. Comparing to the conventionalmethod of directly Fourier transforming the phase masks[2–4], our computer-generated holography technique[11,17] is more feasible for constructing the NABs due

to the fine features in the phase gradient [see Figs. 1(c) and2(b)] and the ability to encode the intensity information[17]. Our experimental results are shown in Fig. 3, where(a)–(e) are the direct top-view photographs if the beampropagation is extrapolated from scattered light, and (f)and (g) display the snapshots of the transverse patternstogether with the intensity profiles (along the x direction)taken at z ¼ 0 and z ¼ 50 �m as marked by the dashedlines in (d) and (e). From these figures, it can be seen thatdifferent elliptical or circular beam paths can be prede-signed by employing different holographic masks accord-ing to Eqs. (4) and (6), in agreement with our theoreticalanalysis of the MABs shown in Fig. 1 but atm ¼ 3132 andq ¼ 2:6� 106. The results in Figs. 3(d)–3(g), correspond-ing to Figs. 2(c) and 2(d), clearly illustrate that the WAB( ¼ 1200) preserves its structure much better than theAiry beam does.Finally, we demonstrate the self-healing property of our

MABs and WABs—an interesting feature typical of non-diffracting self-accelerating wave packets [3,4,10]. Figure 4depicts the numerical and experimental results of the side-view propagation when a MAB or a WAB encounters anobstacle along its trajectory. In these cases, the obstaclepartially blocks the accelerating beams around the mainlobe. As expected, the MABs and WABs restore their beamstructures during subsequent propagation, as if they couldovercome the obstacle. The self-healing behaviors of theMABs and WABs indicate the caustic nature of the beamstructures, which might provide more controllability fromthe point of view of the nonparaxial caustics [9,18].To summarize, we have proposed and demonstrated

new families of nonparaxial self-accelerating beams, lead-ing to a novel, complete picture in terms of acceleratingbeams. In particular, we show that, by designing the MABsand WABs, optical beams with large-angle self-bending,

FIG. 3 (color online). Experimental side-view photography ofthe generated MABs at a < b (a), a ¼ b (b), and a > b (c), aWAB (d) and an Airy beam (e) taken from scattered light, wherethe dotted curves plot the predesigned trajectories. (f) and (g) arethe transverse intensity distributions recorded at z ¼ 0 (f) andz ¼ 50 �m (g) as marked by the dashed lines in (d) and (e),where the top and middle rows correspond to the WAB and Airybeam, respectively, and the bottom depicts the intensity profilestaken along the dashed lines (marked in the transverse intensitypatterns). The dashed (red) and solid (blue) lines correspond tothe Airy beam and the WAB, respectively.

FIG. 2 (color online). Amplitude at z ¼ 0 (a) and phase of theFourier spectra (b) of a Weber accelerating beam (WAB) (blue,solid) and an Airy beam (red, dotted); (c) and (d) depict theside-view propagation of the WAB (c) and the Airy beam(d) corresponding to the profiles in (a), where the white dashedline marks the z ¼ 0 position; (e) illustrates the FWHM of themain lobe of the beam versus the propagation length z fordifferent beams with the same main lobe size at input.

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nondiffracting, and self-healing features along designedcurved trajectories can be achieved, well beyond the para-xial condition required for the Airy beams. One can directlyapply these beams in a variety of applications such aslight-induced plasma channels [6], microparticle manipu-lations [5], and surface plasmon routing in the generalnonparaxial geometry [7]. In addition, the results presentedhere can be easily extended to other linear wave systemsin nature. This approach opens a new door for exploringaccelerating wave packets in the nonparaxial regime.Specifically, this raises many intriguing fundamental ques-tions. For instance, could theMABs survive in the presenceof nonlinearities [19]? If so, is it possible to realize anoptical boomerang traveling along different closed trajec-tories? Since the results directly stem from the Maxwellequations, will such sharply bending NABs bring aboutnew spin-orbital interaction dynamics [20]? In addition, weexpect that many exciting results are yet to come when thisintriguing concept is applied to the time domain [21] orbeyond the diffraction limit [22].

This work was supported by the U.S. ARO MURIprogram (W911NF-09-1-0539) and the NSF NanoscaleScience and Engineering Center (CMMI-0751621). Z.C.was supported by AFOSR (FA9550-12-1-0111) and NSF(PHY-1100842). The research in Canada was supported byNSERC and FQRNT.We thank A. Salandrino for assistance.

*Corresponding author.xiang@berkeley.edu

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FIG. 4 (color online). Numerical simulations (top) and experi-mental results (bottom) of the self-healing properties of a MAB(left) and a WAB (right). (a) and (b) are for beams correspondingto Figs. 1(d) and 2(c), and (c) and (d) correspond to Figs. 3(a)and 3(d), respectively. The main lobe of the beam is blockedby an opaque obstacle (white dot) at a certain propagationdistance. The diameter of the white dots in (a) and (b) is 1 �m,and that in (c) and (d) is about 50 �m.

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